Notice that, while the terminology “generalized cohomology” is standard in algebraic topology with an eye towards stable homotopy theory, it is somewhat unfortunate in that there are various other and further generalizations of the axioms that all still deserve to be and are called “cohomology”. For instance dropping the suspension axiom leads to nonabelian cohomology and dropping the “homotopy axiom” (and taking the domain spaces to be smooth manifolds) leads to the further generality of differential cohomology. This entry here is concerned with the generalization obtained from the Eilenberg-Steenrod axioms by just discarding the dimension axiom. For lack of a better term, we say “generalized (Eilenberg-Steenrod) cohomology” here.

There are functors taking any reduced cohomology theory to an unreduced one, and vice versa. When some fine detail in the axioms is suitably set up, then this establishes an equivalence between reduced and unreduced generalized cohomology:

The fine detail in the axioms that makes this work is such as to ensure that a cohomology theory is a functor on the opposite of the (pointed/pairwise) classical homotopy category. Since this has different presentations, there are corresponding different versions of suitable axioms:

On the one hand, Ho(TopQuillen)Ho(Top_{Quillen}) may be presented by topological spaces homeomorphic to CW-complexes and with homotopy equivalence-classes of continuous functions between them, and accordingly a generalized cohomology theory may be taken to be a funtor on (pointed/pairs of) CW-complexes invariant under homotopy equivalence.

On the other hand, Ho(TopQuillen)Ho(Top_{Quillen}) may be presented by all topological spaces with weak homotopy equivalencesinverted, and accordingly a generalized cohomology theory may be taken to be a functor on all (pointed/pairs of) topological spaces that sends weak homotopy equivalences to isomorphisms.

Notice however that “classical homotopy category” is already ambiguous. Pre Quillen this was the category of all topological spaces with homotopy equivalence classes of maps between them, and often generalized cohomology functors are defined on this larger category and only restricted to CW-complexes or required to preserve weak homotopy equivalences when need be (e.g. Switzer 75, p.117), such as for establishing the equivalence between reduced and unreduced theories.

Moreover, historically, these conditions have been decomposed in several numbers of ways. Notably (Eilenberg-Steenrod 52) originally listed 7 axioms for unreduced cohomology, more than typically counted today, but their axioms 1 and 2 jointly just said that we have a functor on topological spaces, axiom 3 was the condition for the connecting homomorphism to be a natural transformation, conditions which later (Switzer 75, p. 99,100) were absorbed in the underlying structure.

Finally, following the historical development it is common to state the exactness properties of cohomology functors in terms of mapping cone constructions. These are models for homotopy cofibers, but in general only when some technical conditions are met, such that the underlying topological spaces are CW-complexes.

For these reasons, in the following we stick to two points of views: where we discuss cohomology theories as functors on topological spaces we restrict attention to those homeomorphic to CW-complexes. In a second description we speak fully abstractly about functors on the homotopy category of a given model category of ∞\infty-category.

(homotopy invariance) If f1,f2:X⟶Yf_1,f_2 \colon X \longrightarrow Y are two morphisms of pointed topological spaces such that there is a (base point preserving) homotopyf1≃f2f_1 \simeq f_2 between them, then the induced homomorphisms of abelian groups are equal

Proof

By the defining exactness of E•E^\bullet, def. , and the way this appears in def. , using that σ\sigma is by definition an isomorphism.

Unreduced cohomology

In the following a pair(X,A)(X,A) refers to a subspace inclusion of topological spaces (CW-complexes) A↪XA \hookrightarrow X. Whenever only one space is mentioned, the subspace is assumed to be the empty set(X,∅)(X, \emptyset). Write TopCW↪Top_{CW}^{\hookrightarrow} for the category of such pairs (the full subcategory of the arrow category of TopCWTop_{CW} on the inclusions). We identify TopCW↪TopCW↪Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow} by X↦(X,∅)X \mapsto (X,\emptyset).

between the value of E•E^\bullet on the pair (X,A)(X,A) and its value on the mapping cone of the inclusion, relative to a basepoint.

If moreover A↪XA \hookrightarrow X is (the retract of) a relative cell complex inclusion, then also the morphism in cohomology induced from the quotient map p:(X,A)⟶(X/A,*)p \;\colon\; (X,A)\longrightarrow (X/A, \ast) is an isomorphism:

If A↪XA \hookrightarrow X is a cofibration, then this is a homotopy equivalence since Cone(A)Cone(A) is contractible and since by the dual factorization lemmaX∪Cone(A)→X/AX \cup Cone(A)\to X/A is a weak homotopy equivalence, hence a homotopy equivalence on CW-complexes.

Example

As an important special case of : Let (X,x)(X,x) be a pointedCW-complex. For p:(Cone(X),X)→(ΣX,{x})p\colon (Cone(X), X) \to (\Sigma X,\{x\}) the quotient map from the reduced cone on XX to the reduced suspension, then

Proposition

Given E•E^\bullet an unreduced cohomology theory, def. . Given a topological space covered by the interior of two spaces as X=Int(A)∪Int(B)X = Int(A) \cup Int(B), then for each C⊂A∩BC \subset A \cap B there is a long exact sequence of cohomology groups of the form

where on the right we have, from the construction, the reduced mapping cone of the original inclusion A↪XA \hookrightarrow X with a base point adjoined. That however is isomorphic to the unreduced mapping cone of the original inclusion. With this the natural isomorphism is given by lemma .

where on the right we have the reduced mapping cone of the point inclusion with a point adoined. As before, this is isomorphic to the unreduced mapping cone of the point inclusion. That finally is clearly homotopy equivalent to XX, and so now the natural isomorphism follows with homotopy invariance.

Finally we record the following basic relation between reduced and unreduced cohomology:

Proposition

Let E•E^\bullet be an unreduced cohomology theory, and E˜•\tilde E^\bullet its reduced cohomology theory from def. . For (X,*)(X,\ast) a pointed topological space, then there is an identification